Problem: Find $\dfrac{d}{dx}[15\log(x)]$. Choose 1 answer: Choose 1 answer: (Choice A) A $\dfrac{15}{x\ln(10)}$ (Choice B) B $\dfrac{15}{x\log(x)}$ (Choice C) C $\dfrac{15\ln(10)}{x}$ (Choice D) D $\dfrac{15\log(x)}{x}$
Solution: The expression to differentiate includes a logarithmic term. Remember that the derivative of the general logarithmic term $\log_a(x)$ (where $a$ is any positive constant and $a\neq 1$ ) is $\dfrac{1}{\ln(a)\cdot x}$. Put another way, $\dfrac{d}{dx}[\log_a(x)]=\dfrac{1}{\ln(a)\cdot x}$. [Is there an easy way to memorize that?] We can use this to find the derivative as shown below. $\begin{aligned} &\phantom{=}\dfrac{d}{dx}[15\log(x)] \\\\ &=15\dfrac{d}{dx}[\log(x)] \\\\ &=15\dfrac{d}{dx}[\log_{10}(x)]&&{\gray{\text{Since }\log(x)=\log_{10}(x)}} \\\\ &=15\cdot\dfrac{1}{\ln(10)x} \\\\ &=\dfrac{15}{x\ln(10)} \end{aligned}$ In conclusion, $\dfrac{d}{dx}[15\log(x)]=\dfrac{15}{x\ln(10)}$.